Let be a differentiable manifold. Show that for each the tangent space is a submanifold of .

Let's see some work! You've posted a lot of questions and have yet to really show your work!

Which working definition are you using? Immersion by the inclusion map? Plain old that for each $\displaystyle x\in T_pM$ there is a chart $\displaystyle (U,\varphi)$ such that $\displaystyle x\overset{\varphi}{\mapsto}\bold{0}$ and $\displaystyle \varphi\left(U\cap T_pM\right)=\left\{x\in\varphi(U):\pi_{k+1}(x)=\cd ots=\pi_n(x)=0\right\}$? That last definition only works for a $\displaystyle k$-dimensional submanifold of a $\displaystyle n$-dimensional smooth manifold. Are we working with finite dimensional smooth manifolds?

An -dimensional manifold is a submanifold of another -dimensional manifold () if
a) ( is a subset of )
b) The inclusion map with is an embedding
(Which means that for each the differential is and that the inclusion map is a homomorphism)

So I only care for finite dimensional differential manifolds.

Well I obviously have (a). Could I use the Reverse function therem for the differential?